You know, really, this is a tough thing to answer.
Look at the existing precedents for a 'world union', and you'll see why
The US is a union of states. Ostensibly, they were once seperate countries.
What happened? Well, Washington D.C. seems to have been created specifically to be the capitol.
Australia's states were also seperate colonies at one point.
And Canberra derived from an argument being held between Melbourne and Sydney as to which was more important.
In the end, a new city was created to resolve the argument.
Then there's the EU.
You might notice, that the EU parliament is in Brussels, but the High Court is in Den Haag (The Hague) in the Netherlands.
This might seem arbitrary, but it makes perfect sense when you realise what the EU actually is;
The while the European Union is of course intended to unify Europe as a whole, and officially began as a union of 12 countries, with open borders being the first thing to occur, the EU is actually derived from an older agreement.
In the past, there was an open borders policy between 3 countries, which collectively was known as the Benelux. - In case the name doesn't clear it up, that's Belgium, The Netherlands and Luxembourg.
If you take the Benelux as the starting point of the EU, is it any wonder then that it's major institutions are located in Belgium and the Netherlands?
And therein lies the problem.
Predicting the 'capital' of a united planet would depend to a large extent on how it came into existence.